PHYSICAL REVIEW B 94, 155132 (2016)

Existence of topological nontrivial surface states in strained transition metals: W, Ta, Mo, and Nb

Danny Thonig,1 Tomas Rauch,2 Hossein Mirhosseini,3,* Jurgen Henk,2,† Ingrid Mertig,2,3 Henry Wortelen,4

Bernd Engelkamp,4 Anke B. Schmidt,4 and Markus Donath4

1Department of Physics and Astronomy, Material Theory, University Uppsala, Box 516, 75120 Uppsala, Sweden2Institute of Physics, Martin Luther University Halle-Wittenberg, Von-Seckendorff-Platz 1, 06120 Halle (Saale), Germany

3Max Planck Institute of Microstructure Physics, Weinberg 2, 06120 Halle (Saale), Germany4Physikalisches Institut, Westfalische Wilhelms-Universitat, Wilhelm-Klemm-Strasse 10, 48149 Munster, Germany

(Received 12 May 2016; revised manuscript received 15 July 2016; published 19 October 2016)

We show that a series of transition metals with strained body-centered cubic lattice—W, Ta, Nb, and Mo—hostssurface states that are topologically protected by mirror symmetry and, thus, exhibits nonzero topologicalinvariants. These findings extend the class of topologically nontrivial systems by topological crystalline transitionmetals. The investigation is based on calculations of the electronic structures and of topological invariants. Thesignatures of a Dirac-type surface state in W(110), e.g., the linear dispersion and the spin texture, are verified.To further support our prediction, we investigate Ta(110) both theoretically and experimentally by spin-resolvedinverse photoemission: unoccupied topologically nontrivial surface states are observed.

DOI: 10.1103/PhysRevB.94.155132

I. INTRODUCTION

Topological insulators have become an exciting topic incondensed matter physics [1]. Insulating in the bulk, thesesystems host topologically protected surface states that crossthe global band gap and exhibit spin-momentum locking.These features render them very valuable for fundamentalresearch and promising for spintronic applications. Up tonow, most investigations have addressed strong topologicalinsulators (TIs; in particular strained HgTe [2] as well asBi2Te3 and similar compounds [3]) and topological crystallineinsulators (TCIs, e.g., SnTe [4,5]).

Recent experimental investigations of the surface electronicstructure of W(110) have brought a remarkable surface state toattention [6–8], followed up by theoretical calculations [9–13].This surface state is strongly spin polarized due to Rashba spin-orbit coupling [14–16]. But more strikingly, it shows linear andstrong dispersion along the �-H high-symmetry line of thesurface Brillouin zone [Fig. 1(a)]. It is therefore reminiscentof a TI’s surface state [1,3], although it becomes flattened along�-N due to the twofold rotational symmetry of the surface.

The salient properties of this surface state immediately raisethe question whether it is a “true” topologically nontrivialsurface state (TSS) or it is “only” reminiscent of a TSS.In this paper we prove that the surface state of W(110) isindeed topologically protected; this is achieved by calculatingthe respective topological invariants for slightly strained Wbulk (compressed by up to 4% in the calculations). Ontop of this, we show that it has analogs in other transitionmetals with body-centered-cubic (bcc) lattice: Ta, Nb, andMo. We provide both experimental and theoretical evidencefor the exemplary material Ta. Having identified a numberof transition metals hosting TSSs, our findings call forinvestigating other material classes—besides insulators andsemimetals [17–23]—that may host topologically protected

*Present address: Max Planck Institute for Chemical Physics ofSolids, Nothnitzer Str. 40, 01187 Dresden, Germany.

†Corresponding author: juergen.henk@physik.uni-halle.de

edge states. Recently, Au(111) with increased strength of thespin-orbit coupling has been identified topologically nontrivial[24].

The paper is organized as follows. In Sec. II we presentthe theoretical and experimental methods used to describetopological nontrivial surface states. Results and discussionsare presented in Sec. III. Finally, we give conclusions inSec. IV. Further details are given in the Appendixes.

II. METHODS

Ab initio electronic-structure calculations for both bulk and(110) surfaces of W, Ta, Nb, and Mo have been performedwithin the local density approximation to density-functionaltheory (DFT), using Perdew-Burke-Ernzerhof generalizedgradient exchange-correlation functionals [25,26] and twoindependent methods: Korringa-Kohn-Rostoker and VASP.Experimental investigations of the unoccupied states ofTa(110) were performed by spin- and angle-resolved inversephotoemission.

A. Korringa-Kohn-Rostoker calculations

We have applied relativistic multiple-scattering theory asformulated in the Korringa-Kohn-Rostoker (KKR) approach[27,28] that is based on multiple-scattering theory [29]. Solv-ing the Dirac equation, relativistic effects are fully accountedfor, especially the spin-orbit interaction which is essentialfor heavy elements such as Ta and W. Finite-size effects areexcluded by modeling the surfaces in a semi-infinite geometry.The computations have been performed with the OMNI programpackage [30] which is an implementation of the layer-KKRscheme [27]; this approach is well suited for semi-infinitesystems (e.g., surfaces and interfaces).

The spectral density niα(E,k‖), i.e., the energy- and wave-vector-resolved local density of states for a site α in layer i, iscomputed from the site-resolved Green function Giα,iα(E +iη,k‖),

niα(E,k‖) = − 1

πIm Tr Giα,iα(E + iη,k‖). (1)

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DANNY THONIG et al. PHYSICAL REVIEW B 94, 155132 (2016)

FIG. 1. Surface electronic structure of W(110). (a) Topological surface state (marked TSS). The spectral density of the topmost layersis shown as normalized color scale for the �-H line of the surface Brillouin zone. (b) As (a), but spin resolved with respect to the Rashbacomponent of the spin polarization (±1 is fully spin polarized: −1 spin-down, +1 spin-up). (c) Brillouin zone of a bcc lattice and its projectiononto the (110) surface. Red lines indicate an irreducible part. H and H ′ become inequivalent under strain. (d) Schematic illustration of thesurface-state dispersion from (a), with spin polarization indicated by colors. Recall that bulk states show up also in the dark green area around� because they are projected onto the surface Brillouin zone.

η is a small offset from the energy axis leading to a broadeningof the spectral density; typically η ≈ 0.01 eV. The spectraldensity can further be decomposed with respect to spinpolarization and angular momentum, thus allowing for adetailed characterization of the electronic states.

B. VASP calculations

The KKR calculations are complemented by analogouscomputations with the VASP program package [31,32], usinga slab geometry. The Vienna ab initio simulation package(VASP) solves the Kohn-Sham equations by augmented planewaves basis sets [31,32]; this allows us to treat the full crystalpotential and, consequently, structural relaxations. Relativisticeffects are accounted for by first-order perturbation in the spin-orbit coupling which requires handling of the core states byprojector-augmented-wave pseudopotentials (PAW) [33,34].The Perdew-Burke-Ernzerhof generalized gradient approxi-mation (PBE-GGA) for the exchange correlation avoids sideeffects from strongly varying charge densities in W, Ta, Nb,and Mo. To mimic (110) surfaces, we utilized slabs 15 layersthick in which relaxations are allowed in the first 8 layers.The vacuum region is chosen as wide as 27 bulk interlayerdistances.

The reciprocal space is partitioned using a Monkhorstmesh with 21 × 21 × 21 and 21 × 21 × 1 points for bulk andslab calculations, respectively. Plane wave expansions in thevalence-band region are cutoff at 520 eV.

The electronic structures obtained by the VASP and KKRmethods agree very well, putting our findings on firm ground.

C. Surface relaxations

Surface relaxations have been determined by VASP calcu-lations (Table I; to obtain optimal structural relaxations, weutilized a Methfessel-Paxton smearing of the order 2 with abroadening of 0.05 eV). They are important ingredients in ourreasoning.

Experimental lattice parameters for W(110) agree with thetheoretical predictions: Ref. [37] gives d12 = −2.2 ± 1.0%,whereas Ref. [38] states d12 = −2.7(5)% and d23 < 0.3%.The calculated layer relaxations agree with those of previoustheoretical studies: for Mo(110) [39–42], for Nb(110) [43],and for Ta(110) [44].

D. Tight-binding calculations

The DFT results serve as input for tight-binding (TB)parametrizations for W, Ta, Nb, and Mo, from which wecalculate Berry curvatures and mirror Chern numbers nm.

The TB calculations for bcc W, Ta, Nb, and Mo rely onthe Slater-Koster parametrization [45] including spin-orbitcoupling; the TB parameter are listed in Appendix A. First- andsecond-nearest-neighbor parameters have been considered.For the distorted systems, the TB parameters have been scaledaccording to Harrison [46].

For the surface electronic structure, we have applied Greenfunction renormalization which treats semi-infinite systemsaccurately [47,48]. This approach yields the spectral density,see Eq. (1).

The tight-binding approach proved successful for, e.g.,Bi2Te3, Bi2Se3, SnTe, and HgTexS1−x [49–51]. Recall that

TABLE I. Geometry of bcc(110) surfaces obtained from VASP

calculations. The relative changes of the distances dij between layeri and j is given with respect to the bulk interlayer distance; i,j =1,2,3, . . . indicate the topmost, second, third layer, etc., a is the latticeconstant. Data for Ta reproduced from [35,36].

W Ta Mo Nb

d12 − 3.67% − 4.81% − 4.96% − 3.77%d23 +0.92% +0.57% +1.32% +1.19%d34 +0.20% +0.29% +0.41% +0.11%a 3.172 A 3.308 A 3.151 A 3.323 A

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our TB and DFT calculations are in good agreement(Appendix A).

E. Mirror Chern numbers

A mirror Chern number classifies systems in which thetopological protection is brought about by mirror symmetry inthe bulk (topological crystalline insulators, TCIs). We considermirror planes which are perpendicular to the (110) surfaceand cut the �-H and �-N lines of the surface Brillouin zone[Fig. 1(c)]. The modulus of nm equals the number of TSSsfor the respective mirror plane, its sign determines the spinchirality of these TSSs.

The mirror Chern number nm of the N lowest bands isdefined as [52]

nm = 12 (n+i − n−i), (2)

with

n±i =∫

�±i(k) · en d2k, (3)

the Chern number of the N/2 states with mirror eigenvalue of±i. en is a unit vector normal to the mirror plane and

�±i(k) = i

N2∑

n=1

⟨∇ku±in (k)

∣∣ × ∣∣∇ku±in (k)

⟩(4)

is the sum of the Berry curvatures of all these N/2 bands.u±i

n (k) is the periodic part of the nth Bloch function withmirror eigenvalue ±i. The integration in Eq. (3) is over theintersection of the mirror plane with the bulk Brillouin zone.

If nm �= 0, the bulk-boundary correspondence [1] tells therehave to be |nm| surface states connecting the N th with the(N + 1)th bulk band. These surface states are located at theprojection of the mirror plane onto the surface Brillouin zone.

F. Inverse-photoemission experiments

The Ta(110) surface was cleaned by repeated cycles ofheating. Prolonged heating in an oxygen atmosphere of6 × 10−8 mbar at temperatures of 1800 K was followed byshort-time heating to temperatures of 2700 K. These hightemperatures were necessary to dissolve the strong surfaceinteraction with oxygen. The cleaning procedure was effectiveto remove contaminants, such as C and O, from the surface. Thesurface quality was confirmed by Auger electron spectra andby a (1 × 1) low-energy electron diffraction pattern with sharpdiffraction spots and low background intensity. Photoemissiondata of the occupied surface state just below the Fermi levelserved as an additional sensitive criterion for a clean surface.For details see Refs. [35,53].

The experimental setup is depicted in Fig. 2. The samplewas irradiated with spin-polarized electrons of defined energyand momentum, given by the angle of incidence θ with respectto the surface normal n. The electrons are photoexcited from aGaAs cathode in our rotatable spin-polarized electron sourceROSE [54]. The spin-polarization direction of the impingingelectrons was adjusted perpendicular to their in-plane wavevector k‖ and perpendicular to the surface normal n, i.e., beingsensitive to the Rashba component. The azimuth of the samplecan be adjusted by varying the angle �. The emitted photons

FIG. 2. Geometry for the spin- and angle-resolved inverse-photoemission experiments.

(�ω = 9.9 eV) were detected at an angle of 65 ◦ relative to theelectron beam in the plane of incidence and 32 ◦ perpendicularto it. More information about the experiment is given inRefs. [35,55].

III. RESULTS AND DISCUSSION

A. Semimetal band gap and band inversion

Since the Dirac-type surface state of W(110) has beeninvestigated in every detail [6,7,11,13,56,57], we provideexperimental evidence for TSSs in Ta(110). For this purposewe used spin- and angle-resolved inverse photoemission(IPE). The spin-dependent unoccupied electronic structure ofTa(110) was investigated by utilizing a spin-polarized electronbeam [54] and measuring the Rashba component of the spinpolarization (for details, see Sec. II F).

Topologically nontrivial systems are characterized by bandinversions which give rise to nonzero topological invariants;the latter are defined for and computed from the bulk states.If the system exhibits either a global or a semimetal gap[58] the invariant is integer, which is obviously the casefor insulators. Focusing first on W(110), we are facing twoproblems: semimetal band gap and band inversion.

Semimetal band gap. Bulk W does not have a band gap inthe energy region in which the relevant surface state showsup [Figs. 1(a) and 3(a)]. However, both compressive or tensilestrain in [110] direction open up the desired semimetal gap inthe �-H and �-N mirror planes, thus allowing us to computethe mirror Chern numbers nm. We have applied strain up to±4% which is in the range of the surface relaxation (Table I)and would like to emphasize that any nonzero strain breaksthe point-group symmetry and causes the opening of a gap.Such a strain could be studied by growing metal films on, e.g.,a piezocrystal [59].

Band inversion. Although the Dirac-type surface state isobserved at the � point of the surface Brillouin zone, therelevant band inversion takes place at the H points of the bulkBrillouin zone (Fig. 3), i.e., at energies larger than the Fermilevel EF. The “small group” of H is Oh and the six t2g

bulk bands are split into one twofold degenerate E5/2g andone fourfold degenerate G3/2g level if spin-orbit coupling isconsidered. Under strain along [110], the small group of H

is reduced to D2h. The E5/2g level stays twofold degenerate,the G3/2g level is further split into two twofold degeneratelevels. All these levels belong to the representation E1/2g . Thissplitting shows up for both tensile and compressive strain andis observed in TB as well as KKR (cf. Appendix B).

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(110)

(001)

(110)

a1

a2 a3σ

FIG. 3. Band inversion and band gap opening in W. Tight-binding band structures are shown along those lines in the bulk Brillouin zonethat are relevant for the topological phase transition. (a) Cubic W without (gray) and with (black) spin-orbit coupling. The inset shows a zoomto the topological “hot spot” of the dispersion (b) body-centered cubic crystal, with lattice sites represented by spheres; the green area visualizesthe unit cell of the (110) surface spanned by the lattice vectors a1 and a2. The lattice vector between the (110) planes is a3 which is altered uponstrain σ . The two mirror planes—with surface normal in (001) and (110)—for which the Chern numbers have been computed are displayed inblue and red. (c) Tensile and compressively strained W with spin-orbit coupling. Insets show band dispersions that are relevant for the topologyof W under strain at H (red) and H ′ (blue). H ′ and N ′ are defined in Fig. 1(c).

For the strained bulk systems with semimetal band gapswe compute the mirror Chern numbers nm for both the �-Hand �-N mirror planes. For tensile strain we find nm = 0,indicating a topological trivial system. For compressive strain,nm for the �-N mirror plane vanishes as well; we recall that thesurface state is weakly dispersive along this line [7]. However,for the �-H mirror plane—for which the surface state showslinear dispersion—we compute nm = −2. The results of ourcalculations are summarized in Table II. This finding indicatesthat compressively strained W is a metal hosting TSSs in thesemimetal gap. It further tells us that the semimetal band gaphas to be bridged by two TSSs with identical spin chirality.Analogous calculations for Ta, Mo, and Nb give identicalresults concerning the topological properties. Recall that inW and Mo the TSSs are occupied while in Ta and Nb they areunoccupied.

To provide qualitative insight into the complicated elec-tronic structure we turn to Ta(110) [Fig. 4(a)]. In the TB cal-culations for the surface we assume homogeneously strainedsamples (i.e., without detailed surface relaxation) and bulkTB parameters in the topmost layers. By calculating the bulk

TABLE II. Mirror Chern numbers nm of the �-H and the �-Nplane calculated for both compressive and tensile strain.

�-H �-N

Tensile 0 0Compressive −2 0

band structure along �-H–N for a set of equidistant k⊥, weachieve a representation of the (E,k)-dependent semimetalband gap: the band that forms its lower (upper) boundary iscolored green (red). This band structure is superimposed ontothe surface spectral density which shows two surface states.The surface state TSS1 starts at 1.0 eV at � off a green bulkband and can be traced to H where it snuggles up to bulkband edge; then it disperses to higher energies at N where itconnects to a red bulk band. TSS2 starts at 1.45 eV at � offa red band and reaches a green bulk band close to H . Thespin-resolved spectral density tells that TSS1 and TSS2 haveopposite spin polarization [Fig. 4(b)]. Note that TSS1 (TSS2)exhibits its part above (below) its Dirac point at � which hasopposite spin polarization as compared to its lower (upper)part [cf. Figs. 1(b) and 1(d) for W]. The spin chirality of TSS1and TSS2 is therefore identical, which is in line with the mirrorChern number nm = −2.

B. Topological surface states in Ta(110), Mo(110), and Nb(110)

We now show by ab initio calculations and IPE experimentsthat Ta(110) (cf. Appendix C) also hosts TSSs. Ta lends itselffor an investigation because its strong spin-orbit couplingproduces a sizable spin-orbit band gap. The semimetal gapis between the dispersive bands that become inverted bycompressive strain (cf. Fig. 7). Near � it shows up atEF + 1.1 eV [Fig. 4(c)]. Two surface bands with opposite spinpolarization are split off the bulk band edges at �, one fromthe lower, the other from the upper band edge, in accordance

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FIG. 4. Unoccupied electronic structure of Ta(110) for the �-Hline of the surface Brillouin zone. (a) Spectral density of the topmostsurface layer of unstrained Ta calculated with the tight-bindingmethod. The bulk bands that form the boundary of the (E,k)-dependent band gap are shown in green and red. (b) As (a) but resolvedwith respect to the Rashba spin component. (c) and (d) As (a) and(b) but calculated by the ab initio KKR method. The symbols resultfrom spin-resolved inverse-photoemission experiments and indicatepeak positions derived from spectra shown in Fig. 8. The topologicalsurface states are marked TSS1 and TSS2.

with nm = −2 [Figs. 4(c) and 4(d) as well as Fig. 1(d)]. Thetwo TSSs do not exhibit the typical Rashba-type dispersion, asobserved in Au(111) and Bi/Ag(111) [60–62]. Along the �-Hline they disperse in “unison” with nonlinear dispersion [63].Surface bands appear at lower energies relative to the bulk bandedges than those of W, which is attributed to the larger latticeconstant a and the stronger surface relaxation of Ta comparedto W (Table I). An increased lattice constant “flattens” the bulkbands, resulting in downshifted surface bands and stronger hy-bridizations with the bulk states. Since the TSS in W(110) is lo-cated at the lower boundary of the band gap at � (about −1.25to −0.75 eV in Fig. 1(a) [11]), the respective Ta surface stateand its Dirac point are “hidden” in the bulk bands at � [64,65].

Strained Ta, Nb, and Mo possess the same topologicalinvariants as W. Both Nb and Mo host TSSs as well (Fig. 5; cf.Ref. [40] for Mo). Since both Nb (Z = 41, 4d45s1) and Mo(Z = 42, 4d55s1) are lighter than Ta (Z = 73, 5d36s2) andW (Z = 74, 5d46s2) the band gaps induced by the spin-orbitinteraction are significantly smaller. The surface state in Moresembles a Dirac-like state at � and E − EF = −1.2 eV; thedispersion is not linear, in agreement with experiment [66,67].The orbital composition of the surface states is similar to thosein Ta(110) and W(110).

FIG. 5. Spin-resolved surface electronic structure of Nb(110)(a) and Mo(110) (b), analogous to Fig. 4(d).

IV. CONCLUSION

In summary, we showed that the transition metals W, Ta, Nb,and Mo host topologically protected surface states. Nonzerocompressive strain opens an inverted band gap at the H pointof the bulk Brillouin zone which causes nonzero topologicalinvariants (mirror Chern numbers nm = −2). These findingsindicate the existence of topological surface states withidentical spin polarization at the (110) surfaces. Since the(110) surfaces are effectively compressed (Table I), the fourconsidered bcc metals appear topologically nontrivial althoughtheir bulk lattice is cubic.

To further confirm the predictions for Nb and Mo, weencourage investigations of their electronic structure by spin-resolved conventional or inverse photoemission. The topolog-ical phase transition upon strain could be studied by growingmetal films on either different substrates, on a piezocrystal[59,68], or by bending the sample; strain in the bulk then wouldshow up in addition to the already effectively compressedsurface. Other surfaces are worth investigating as well.

ACKNOWLEDGMENTS

We acknowledge fruitful discussions with Albert Fert. Thiswork is supported by the Priority Program SPP 1666 of DFG.

APPENDIX A: TIGHT-BINDING PARAMETERS

The tight-binding parameters for bcc W, Ta, Nb, andMo (Table III) have been obtained by fitting the bulk bandstructures of the ab initio KKR calculations using a MonteCarlo approach, with focusing on agreement in the energyregion of the topologically nontrivial surface states. Theagreement of the TB and the DFT band structure is visualizedfor W in Fig. 6.

APPENDIX B: STRAIN IN [110] DIRECTIONAND BAND INVERSION

Topological invariants are defined for groups of bands thatare separated by band gaps. More precisely, to calculate themirror Chern number nm for the lowest N bands, there has to

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DANNY THONIG et al. PHYSICAL REVIEW B 94, 155132 (2016)

TABLE III. Slater-Koster tight-binding parameters for W, Ta,Nb, and Mo in units of eV. Es , Ep , Ed , λp , and λd stand forthe on-site energies and spin-orbit coupling constants, respectively.The subscripts indicate the neighbors’ order (1 for first- and 2 forsecond-nearest neighbors.)

W Ta Mo Nb

Es 12.532 11.232 14.997 12.741Ep 17.839 16.648 18.517 16.223Ed 7.964 8.140 7.400 7.083λp 0.86 0.65 0.24 0.19λd 0.18 0.137 0.05 0.043

(ssσ )1 − 1.574 − 1.405 − 1.623 − 1.455(spσ )1 2.486 2.414 2.226 − 2.105(sdσ )1 − 1.582 − 1.542 1.035 1.251(ppσ )1 2.429 2.270 2.495 2.260(ppπ )1 − 0.536 − 0.539 − 0.535 − 0.503(pdσ )1 − 2.188 − 2.128 1.874 − 1.768(pdπ )1 0.028 0.068 0.009 − 0.032(ddσ )1 − 1.573 − 1.561 − 1.404 − 1.386(ddπ )1 0.743 0.731 0.666 0.657(ddδ)1 − 0.023 − 0.048 − 0.052 − 0.007

(ssσ )2 − 0.291 − 0.302 − 0.355 − 0.203(spσ )2 0.508 0.361 0.681 − 0.580(sdσ )2 − 0.831 − 0.563 1.018 0.738(ppσ )2 0.891 0.857 1.006 0.936(ppπ )2 − 0.056 − 0.066 − 0.043 − 0.114(pdσ )2 − 1.613 − 1.649 1.546 − 1.481(pdπ )2 0.247 0.242 − 0.374 0.227(ddσ )2 − 0.908 − 0.838 − 0.739 − 0.852(ddπ )2 0.188 0.245 0.280 0.223(ddδ)2 0.086 − 0.007 0.006 0.080

FIG. 6. Bulk band structure of cubic W without spin-orbitcoupling calculated with VASP (black) and with the tight-bindingapproach (red). EF is the Fermi energy.

FIG. 7. Bulk band inversion in W, obtained from KKR (left panel)and TB (right panel). The band inversion is mediated by strain of −3%(compressive strain; black line), 0% (red line), and +3% (tensilestrain; blue line). Arrows and dashed lines indicate the band gapinversion associated with compressive strain.

be a finite gap between the N th and (N + 1)th band for allwave vectors in the considered mirror plane.

In the case of W, the Dirac-type surface state shows up inthe local gap at �, which opens up between bands 6 and 7,both of which belong to t2g orbitals. Considering both mirrorplanes (�-H and �-N ), one finds that this gap closes at theH point, ending in a fourfold degenerate level. Lowering thesymmetry by applying strain in [110] direction, this fourfolddegenerate level splits into two twofold degenerate ones. Forboth compressive and tensile strain, bands 6 and 7 remain

E-E (eV)F

3210E-E (eV)F

3210

θHΓ

(a)

20°

25°

30°

35°

40°

20°

25°

30°

35°

40°

θHΓ

(b)

TSS1 TSS2

)stinu .br a( yt isnetnI

FIG. 8. Spin-integrated (a) and spin-resolved (b) IPE spectra forTa(110) for various angles of incidence θ in the � H mirror-symmetrydirection. In (b), spin-up (spin-down) is marked by red (blue)triangles. The shape of the background intensity is indicated by solidlines below the surface-state emissions. The precise procedure fordetermining the energy positions of TSS1 and TSS2 is described indetail in Ref. [35].

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separated in the entire �-H and �-N mirror planes, allowingus to calculate nm for both planes.

To further support the TB approach, we compare the KKRwith the TB electronic structure in the vicinity of the H point ofthe bulk Brillouin zone (Fig. 7). We applied 3% of tensile andcompressive strain in [110] direction. For the strained systemsa band gap opens up at H : the degeneracy of the bands of thecubic system (red) is lifted for the strained systems (black andblue). But only compressive strain leads to a band inversion:due to spin-orbit coupling a band crossing is avoided, as isindicated by dashed lines and arrows. This band inversion isthe origin for nonzero mirror Chern numbers in W, Ta, Mo,and Nb.

APPENDIX C: IPE SPECTRA FOR TA(110)

Spin-integrated and spin-resolved IPE spectra along � H

are shown in Fig. 8 for θ between 20◦ and 40◦. They revealthe dispersion and spin texture of TSS1 and TSS2. Fordetermining of the E(k‖) dispersion, the nearby bulk-bandedge has to be taken into account. Therefore, the spectrawere decomposed into a spin-polarized surface and a spin-independent bulk contribution on top of a spin-independentbackground intensity. For details, see Ref. [35]. The results ofthis fitting routine are included as red (TSS1) and blue dots(TSS2) in Fig. 4(d) of the paper in direct comparison with thecalculated energy dispersion.

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